The Decimation Method for Laplacians on Fractals: Spectra and Complex Dynamics

TL;DR

This survey explores the decimation method for analyzing the spectra of fractal differential operators, highlighting its application to self-similar fractals like the Sierpinski gasket and its role in understanding spectral zeta functions.

Contribution

It generalizes the decimation method to various fractals using complex rational maps and links the dynamics of these maps to spectral properties and zeta functions.

Findings

Decimation method effectively describes spectra of fractal operators.

Renormalization maps are polynomial or rational functions depending on fractal type.

Dynamics of these maps are crucial for spectral analysis and zeta function factorization.

Abstract

In this survey article, we investigate the spectral properties of fractal differential operators on self-similar fractals. In particular, we discuss the decimation method, which introduces a renormalization map whose dynamics describes the spectrum of the operator. In the case of the bounded Sierpinski gasket, the renormalization map is a polynomial of one variable on the complex plane. The decimation method has been generalized by C. Sabot to other fractals with blow-ups and the resulting associated renormalization map is then a multi-variable rational function on a complex projective space. Furthermore, the dynamics associated with the iteration of the renormalization map plays a key role in obtaining a suitable factorization of the spectral zeta function of fractal differential operators. In this context, we discuss the works of A. Teplyaev and of the authors regarding the examples of the bounded and unbounded Sierpinski gaskets as well as of fractal Sturm-Liouville differential operators on the half-line.